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Chapter 4: Problem 53

Concentrated stop bath for photography is \(28 \%\) acetic acid. Find the amount of concentrated stop bath and the amount of water needed to make \(3 \mathrm{~L}\) of a new mixture that is \(2 \%\) acetic acid. Round to the nearest tenth.

Short Answer

Expert verified
0.2 liters of concentrated stop bath and 2.8 liters of water.

Step by step solution

01

Define the Variables

Let the volume of the concentrated stop bath (which is 28% acetic acid) be denoted as \(x\) liters. The volume of water added will be \(3 - x\) liters, since the total volume of the mixture is 3 liters.
02

Set Up the Equation

The amount of acetic acid from the stop bath is \(0.28x\) and the amount of acetic acid in the final mixture is \(0.02 \times 3 = 0.06\) liters. Set up the equation: \(0.28x = 0.06\).
03

Solve for x

Solve the equation for \(x\): \[0.28x = 0.06\] Divide both sides by 0.28: \[x = \frac{0.06}{0.28} \approx 0.214\] liters. Round this to the nearest tenth: \[x \approx 0.2\] liters.
04

Calculate the Amount of Water Needed

The amount of water needed is \(3 - x\) liters: \[3 - 0.2 = 2.8\] liters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Concentration
In a mixture problem, concentration refers to the amount of a specific substance within a certain volume of a solution. For instance, in the given exercise, we are dealing with a 28% acetic acid solution. Here, '28%' indicates the concentration, meaning 28% of the solution鈥檚 volume is acetic acid. When we dilute this with water, the concentration changes. Understanding concentrations helps us determine how much of each component to mix to achieve the desired new concentration. It鈥檚 essential to set up your problem correctly and define variables that express the volumes of the original and diluted solutions.
Linear Equations
Linear equations are a powerful tool in solving mixture problems. These equations help us lay out the relationships between known and unknown quantities. In our exercise, we created a linear equation to represent the total amount of acetic acid in the final solution. We did this by multiplying the volume of each component by its concentration: \[volume * concentration = total\rightarrow 0.28x + 0 = 0.06\rightarrow 0.28x = 0.06 \]. When we solve this equation, we find the volume (\
Problem Solving
Solving mixture problems involves several key steps. First, identify what you know and what you need to find out. For our exercise:
  • We know we need 3 liters of a 2% acetic acid solution.
  • We need to determine how much 28% acetic acid and how much water should be used.
Next, define variables and set up equations that relate these variables. By carefully organizing information and systematically following the equation, you can logically solve for the unknowns. Remember always to double-check your work!
Proportions
Proportions play a crucial role in mixture problems. They allow us to compare the parts of the solution relative to the whole. In this case, we are balancing the proportions of acetic acid and water to achieve a target concentration. The value found for x in our exercise is a fraction of the desired total volume. \(\frac{0.06}{0.28} = \approx 0.214\) tells us the part of the 28% solution needed to reach the new 2% concentration within 3 liters. Concepts of proportions work hand in hand with linear equations, ensuring we maintain the correct relationship between different components of our mixtures.

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